As already mentioned, the weak coupling RG in 1d is quite different than for \(d> 1\) . However, in special cases, such as for the case of quadratic band-touching in 2d, the RG equations have similar form as in one dimensionThis also applies for quadratic band-touching in 3d with appropriate long-range forces [1]}, [2]}. [3]}, [4]}. As in the usual FL, in these cases there are a number of interactions - represented by a set of running coupling constants, \(g_a(t)\) , that are marginal (dimensionless) by power counting. To lowest non-trivial order, this leads to a perturbative expression for the RG flows (analogous to that in Eq. REF ) of the form
\({dg^{\vphantom{\prime }}_a\over dt} = \Gamma _a^{b,c}\, g^{\vphantom{\prime }}_b\,g^{\vphantom{\prime }}_c + \ldots \quad ,\) 
